Exercise Lecture 7 Solution
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Problem Set 2 Health Economics Spring 2014 An individual with the concave utility function u ( y) has a gr oss inco me of Y . In case of illne ss, treatment costs are L. The proba bility of illness is π 0 , but by spending V units on prevention, the individual can lower it to π 1 . All insurance contracts are fair. (a) Assume that prevention activity is observable and that there are two types of insurance contracts ava ila bl e. If the ind ivi dual exert s pr ev ention ef fort, it pays the pr emium P 1 ( I 1 ) = π 1 I 1 where I 1 is the chosen coverage. If the individual does not engage in prevention, the premium i s P 0 ( I 0 ) = π 0 I 0 where I 0 is the chosen coverage. (i) Write down the expected utility (as a function of coverage I) with and without pre- vention. With prevention: EU 1 ( I ) = (1 − π 1 )u( Y − π 1 I −V ) + π 1 u( Y − π 1 I −V − L + I ) Without prevention: EU 0 ( I ) = ( 1 − π 0 )u( Y − π 0 I ) + π 0 u( Y − π 0 I − L + I ) (ii) Show that the individual will choose full coverage (i.e., I ∗ = L) with and without prevention. Take first-order condition with respect to I . With prevention: ∂ EU 1 ( I ) ∂ I =(1 − π 1 )u ( Y − π 1 I − V )(−π 1 ) + π 1 u ( Y − π 1 I − V − L + I )(1 − π 1 ) = 0 ⇔ u ( Y − π 1 I −V ) = u ( Y − π 1 I − V − L + I ) ⇔ Y − π 1 I −V = Y − π 1 I − V − L + I ⇔ 0 = − L + I ⇔ I ∗ 1 = L. Similarly without prevention: ∂ EU 0 ( I ) ∂ I =(1 − π 0 )u ( Y − π 0 I )(−π 0 ) + π 0 u( Y − π 0 I − L + I )(1 − π 0 ) = 0 ⇔ Y − π 0 I = Y − π 0 I − L + I ⇔ 0 = − L + I ⇔ I ∗ 0 = L. (iii) Under which condition will the individ ual exert preven tion effort in equilibr ium? Utility with prevention: EU 1 ( I ∗ 1 = L) =(1 − π 1 )u( Y − π 1 L −V ) + π 1 u( Y − π 1 L −V − L + L) =u( Y − π 1 L −V ) 1
Transcript of Exercise Lecture 7 Solution
8/12/2019 Exercise Lecture 7 Solution
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Problem Set 2 Health Economics Spring 2014
Utility without prevention:
EU 0( I ∗0 = L) =( 1 − π 0)u(Y − π 0 L) + π 0u(Y − π 0 L− L+ L)
= u(Y − π 0 L)
The individual will exert prevention effort if
EU 1( I ∗1 = L) = u(Y − π 1 L− V ) > u(Y − π 0 L) = EU 0( I ∗0 = L)
or
Y − π 1 L− V > Y − π 0 L ⇔ (π 0 − π 1) L > V
(iv) Interpret this condition.
Gain from prevention (drop in probability of being sick times loss when sick) has tobe larger than cost of prevention.
(b) Assume now that prevention activity is not observable. We want to implement the second-
best contract with (partial) coverage I at a premium P = π 1 I where the individual volun-
tarily engages in prevention activity.
(i) Write down the expected utility (as a function of coverage I) with and without pre-
vention.
With prevention:
EU 1( I ) = ( 1 − π 1)u(Y − π 1 I − V ) + π 1u(Y − π 1 I − V − L+ I )
Without prevention:
EU 0( I ) = ( 1 − π 0)u(Y − π 1 I ) + π 0u(Y − π 1 I − L+ I )
Use the following parameter values: Y = 10 , L = 7 , π0 = 0. 4 , π
1 = 0. 2 and V = 1. For
simplicity, assume now linear utility: u ( y) = y.
(ii) Derive the maximum coverage ˜ I under which the individual voluntarily engages in
prevention effort.
Expected utility with prevention:
EU 1( I ) = 0 . 8(10 − 0 . 2 I − 1) + 0 . 2(10 − 0 . 2 I − 1 − 7 + I )
= 7 . 2 − 0 . 16 I + 0 . 4 − 0 . 04 I + 0 . 2 I = 7. 6
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Problem Set 2 Health Economics Spring 2014
Expected utility without prevention:
EU 0( I ) = 0 . 6(10 − 0 . 2 I ) + 0 . 4(10 − 0 . 2 I − 7 + I )
= 6 − 0 . 12 I + 1 . 2 − 0 . 08 I + 0 . 4 I = 7. 2 + 0 . 2 I
Maximum coverage ˜ I under which the individual voluntarily engages in prevention
EU 1( I ) = 7. 6 ≥ 7 . 2 + 0 . 2 I = EU 0( I ) ⇔ I ≤ 2 ⇔ ˜ I = 2
(iii) Show that this contract is preferred to a contract with full coverage and no preven-
tion. Utility under a contract with prevention and coverage ˜ I :
EU 1( ˜ I ) = 7. 6
Utility under a contract with no prevention and full coverage
EU 0( L) = u(Y − π 0 L) = 10 − 0 . 4∗7 = 7. 2
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